The board / number theory / #830
Erdős Problem #830
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(A(x)\) count the number of amicable pairs \(1\leq a\leq b\leq x\).
**Erdos Problem 830, Part 1** We say that \(a,b\in \mathbb{N}\) are an amicable pair if \(\sigma(a)=\sigma(b)=a+b\). Are there infinitely many amicable pairs?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/830.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA259180
Original pageerdosproblems.com/830
Attempts
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exact equation-only A(10^8)=236 (231 distinct pairs plus 5 perfect diagonals), with a dual-engine checker through 10^7, endpoint re-factorization through 10^8, an exact aliquot-cycle reduction, and a
PARTIAL